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<li><a class="reference internal" href="#">Compare cross decomposition methods</a></li>
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  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-cross-decomposition-plot-compare-cross-decomposition-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="compare-cross-decomposition-methods">
<span id="sphx-glr-auto-examples-cross-decomposition-plot-compare-cross-decomposition-py"></span><h1>Compare cross decomposition methods<a class="headerlink" href="#compare-cross-decomposition-methods" title="Permalink to this headline">¶</a></h1>
<p>Simple usage of various cross decomposition algorithms:
- PLSCanonical
- PLSRegression, with multivariate response, a.k.a. PLS2
- PLSRegression, with univariate response, a.k.a. PLS1
- CCA</p>
<p>Given 2 multivariate covarying two-dimensional datasets, X, and Y,
PLS extracts the ‘directions of covariance’, i.e. the components of each
datasets that explain the most shared variance between both datasets.
This is apparent on the <strong>scatterplot matrix</strong> display: components 1 in
dataset X and dataset Y are maximally correlated (points lie around the
first diagonal). This is also true for components 2 in both dataset,
however, the correlation across datasets for different components is
weak: the point cloud is very spherical.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">sklearn.cross_decomposition</span> <span class="kn">import</span> <span class="n">PLSCanonical</span><span class="p">,</span> <span class="n">PLSRegression</span><span class="p">,</span> <span class="n">CCA</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># Dataset based latent variables model</span>

<span class="n">n</span> <span class="o">=</span> <span class="mi">500</span>
<span class="c1"># 2 latents vars:</span>
<span class="n">l1</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="n">l2</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>

<span class="n">latents</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">l1</span><span class="p">,</span> <span class="n">l1</span><span class="p">,</span> <span class="n">l2</span><span class="p">,</span> <span class="n">l2</span><span class="p">])</span><span class="o">.</span><span class="n">T</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">latents</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">4</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">latents</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="mi">4</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>

<span class="n">X_train</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">Y_train</span> <span class="o">=</span> <span class="n">Y</span><span class="p">[:</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">X_test</span> <span class="o">=</span> <span class="n">X</span><span class="p">[</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:]</span>
<span class="n">Y_test</span> <span class="o">=</span> <span class="n">Y</span><span class="p">[</span><span class="n">n</span> <span class="o">//</span> <span class="mi">2</span><span class="p">:]</span>

<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Corr(X)&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">),</span> <span class="mi">2</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Corr(Y)&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">Y</span><span class="o">.</span><span class="n">T</span><span class="p">),</span> <span class="mi">2</span><span class="p">))</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># Canonical (symmetric) PLS</span>

<span class="c1"># Transform data</span>
<span class="c1"># ~~~~~~~~~~~~~~</span>
<span class="n">plsca</span> <span class="o">=</span> <span class="n">PLSCanonical</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">plsca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">X_train_r</span><span class="p">,</span> <span class="n">Y_train_r</span> <span class="o">=</span> <span class="n">plsca</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">X_test_r</span><span class="p">,</span> <span class="n">Y_test_r</span> <span class="o">=</span> <span class="n">plsca</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">)</span>

<span class="c1"># Scatter plot of scores</span>
<span class="c1"># ~~~~~~~~~~~~~~~~~~~~~~</span>
<span class="c1"># 1) On diagonal plot X vs Y scores on each components</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">221</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_train_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">Y_train_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;train&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">25</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;test&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">25</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x scores&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y scores&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Comp. 1: X vs Y (test corr = </span><span class="si">%.2f</span><span class="s1">)&#39;</span> <span class="o">%</span>
          <span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">224</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_train_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">Y_train_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;train&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">25</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;test&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;o&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">25</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;x scores&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;y scores&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Comp. 2: X vs Y (test corr = </span><span class="si">%.2f</span><span class="s1">)&#39;</span> <span class="o">%</span>
          <span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>

<span class="c1"># 2) Off diagonal plot components 1 vs 2 for X and Y</span>
<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">222</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_train_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_train_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;train&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;*&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;test&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;*&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;X comp. 1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;X comp. 2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;X comp. 1 vs X comp. 2 (test corr = </span><span class="si">%.2f</span><span class="s1">)&#39;</span>
          <span class="o">%</span> <span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">X_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">223</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">Y_train_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">Y_train_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;train&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;*&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;test&quot;</span><span class="p">,</span>
            <span class="n">marker</span><span class="o">=</span><span class="s2">&quot;*&quot;</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">s</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">&quot;Y comp. 1&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">&quot;Y comp. 2&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s1">&#39;Y comp. 1 vs Y comp. 2 , (test corr = </span><span class="si">%.2f</span><span class="s1">)&#39;</span>
          <span class="o">%</span> <span class="n">np</span><span class="o">.</span><span class="n">corrcoef</span><span class="p">(</span><span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">Y_test_r</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">])[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(())</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># PLS regression, with multivariate response, a.k.a. PLS2</span>

<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">q</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">p</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span> <span class="o">*</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
<span class="n">B</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span> <span class="o">+</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)]</span> <span class="o">*</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
<span class="c1"># each Yj = 1*X1 + 2*X2 + noize</span>
<span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">B</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span> <span class="o">*</span> <span class="n">q</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">q</span><span class="p">))</span> <span class="o">+</span> <span class="mi">5</span>

<span class="n">pls2</span> <span class="o">=</span> <span class="n">PLSRegression</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">pls2</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;True B (such that: Y = XB + Err)&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">B</span><span class="p">)</span>
<span class="c1"># compare pls2.coef_ with B</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Estimated B&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">pls2</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>
<span class="n">pls2</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

<span class="c1"># PLS regression, with univariate response, a.k.a. PLS1</span>

<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">p</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span> <span class="o">*</span> <span class="n">p</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">))</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">+</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">X</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span> <span class="o">*</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="mi">5</span>
<span class="n">pls1</span> <span class="o">=</span> <span class="n">PLSRegression</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">pls1</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="c1"># note that the number of components exceeds 1 (the dimension of y)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Estimated betas&quot;</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="n">pls1</span><span class="o">.</span><span class="n">coef_</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>

<span class="c1"># #############################################################################</span>
<span class="c1"># CCA (PLS mode B with symmetric deflation)</span>

<span class="n">cca</span> <span class="o">=</span> <span class="n">CCA</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">cca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">X_train_r</span><span class="p">,</span> <span class="n">Y_train_r</span> <span class="o">=</span> <span class="n">cca</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">Y_train</span><span class="p">)</span>
<span class="n">X_test_r</span><span class="p">,</span> <span class="n">Y_test_r</span> <span class="o">=</span> <span class="n">cca</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">Y_test</span><span class="p">)</span>
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